Problem 116 - University Physics with Modern Physics

Question

Basal Metabolic Rate. The basal metabolic rate is the rate at which energy is produced in the body when a person is at rest. \(A 75-\mathrm{kg}(165-\mathrm{lb})\) person of height 1.83 \(\mathrm{m}(6 \mathrm{ft})\) has a body surface area of approximately 2.0 \(\mathrm{m}^{2}\) . (a) What is the net amount of heat this person could radiate per second into a room at \(18^{\circ} \mathrm{C}\) (about \(65^{\circ} \mathrm{F}\) ) if his skin's surface temperature is \(30^{\circ} \mathrm{C} ?\) (At such temperatures, nearly all the heat is infrared radiation, for which the body's emissivity is \(1.0,\) regardless of the amount of pigment.) (b) Normally, 80\(\%\) of the energy produced by metabolism goes into heat, while the rest goes into things like pumping blood and repairing cells. Also normally, a person at rest can get rid of this excess heat just through radiation. Use your answer to part (a) to find this person's basal metabolic rate.

Step-by-Step Solution

Answer

The net radiated heat is 141.41 W and the basal metabolic rate is 176.76 W, approximately 4056 kcal/day.

Step 1: Understand the Problem

We need to find the heat radiated by a person and use it to determine the basal metabolic rate (BMR). The key informative formula here is the Stefan-Boltzmann Law, which calculates the power radiated by a body in terms of the emissivity, surface area, and temperature difference between the body and environment.

Step 2: Set Up the Stefan-Boltzmann Law

The formula to find the amount of heat radiated is given by \[ P = \, \sigma \cdot A \cdot e \cdot (T_{skin}^{4} - T_{room}^{4}) \]where \( P \) is the power radiated, \( \sigma = 5.67 \times 10^{-8} \, \mathrm{W/m^2 \cdot K^4}\) is the Stefan-Boltzmann constant, \( A = 2.0 \, \mathrm{m^2} \) is the surface area, \( e = 1.0 \) is the emissivity, \( T_{skin} = 303 \, \mathrm{K} \) is the skin temperature in Kelvin, and \( T_{room} = 291 \, \mathrm{K} \) is the room temperature in Kelvin.

Step 3: Calculate the Power Radiated

Substitute the values into the Stefan-Boltzmann equation:\[ P = 5.67 \times 10^{-8} \, \mathrm{W/m^2 \cdot K^4} \times 2.0 \times 1.0 \times (303^4 - 291^4) \]Calculate the individual temperatures raised to the fourth power and subtract:\[ 303^4 = 84197481 \, \text{K}^4 \]\[ 291^4 = 71740961 \, \text{K}^4 \]\[ P = 5.67 \times 10^{-8} \times 2.0 \times (84197481 - 71740961) \]\[ P = 5.67 \times 10^{-8} \times 2.0 \times 12456520 \]\[ P \approx 141.41 \, \mathrm{W} \]

Step 4: Find Basal Metabolic Rate (BMR)

Since 80% of BMR is radiated as heat, and we know this person could radiate approximately 141.41 W, this value represents 80% of the BMR. Set up the equation:\[ 0.80 \times \text{BMR} = 141.41 \, \mathrm{W} \]Solve for BMR:\[ \text{BMR} = \frac{141.41}{0.80} \]\[ \text{BMR} \approx 176.76 \, \mathrm{W} \]

Step 5: Round Off the BMR

BMR is often given as kilocalories per day instead of watts, so we can convert:\[ \text{1 W} = 0.239 \text{ kcal/h} \] and\[ \text{BMR} (\text{kcal/day}) = 176.76 \times 0.239 \times 24 \approx 4056 \text{ kcal/day} \]

Key Concepts

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is essential in understanding how heat radiates from a body. It states that the power radiated per unit area of a black body is proportional to the fourth power of the black body's absolute temperature. In simple terms, this means the hotter an object is, the more heat it emits. The formula used to express this concept is

• \( P = \, \sigma \cdot A \cdot e \cdot (T_{skin}^{4} - T_{room}^{4}) \)
• Where \( P \) is the power radiated, \( \sigma \) is the Stefan-Boltzmann constant, \( A \) is the surface area, \( e \) is the emissivity, \( T_{skin} \) is the skin temperature, and \( T_{room} \) is the room temperature.

The Stefan-Boltzmann constant \( \sigma \) has a value of approx. \( 5.67 \times 10^{-8} \, \mathrm{W/m^2 \cdot K^4} \). Calculations using this law often require temperatures to be in Kelvin for accuracy.

Emissivity

Emissivity is a measure of an object's ability to emit infrared energy. It is a ratio, with values ranging between 0 and 1. An emissivity of 1 indicates a perfect emitter, also known as a blackbody, which is an idealized physical body that absorbs all incident electromagnetic radiation. In most real-world applications, objects have an emissivity less than 1.

• For the case of human skin, the emissivity is often assumed to be close to 1 (0.98-1.0) which behaves nearly like a blackbody at temperature ranges pertinent to human physiology.
• Highly polished metals have much lower emissivity values, reflecting more radiation than they emit.

Understanding emissivity is crucial in accurately calculating the radiation emitted by an object, such as human skin in the context of basal metabolic rate.

Infrared Radiation

Infrared radiation is a type of electromagnetic radiation, just like visible light, but with longer wavelengths. It is essential in understanding how heat is transferred from one body to another without physical contact. Most creatures, including humans, radiate infrared radiation which can be detected through thermal imaging cameras. Infrared radiation is crucial in:

• Night vision technology, which detects heat rather than visible light.
• Climate and weather research, where satellites measure earth’s infrared radiation to estimate temperature.
• Medicine and security, to detect heat patterns in the human body or objects.

In cases where basal metabolic activities are analyzed, nearly all human body heat is emitted as infrared radiation, especially when at rest.

Metabolism

Metabolism refers to the chemical reactions within cells that transform food into energy needed to sustain life processes. This energy is necessary for everything from cellular repair to maintaining body temperature and supporting physical activities. When at rest, a significant portion of energy produced by metabolism is released as heat, primarily through radiation.

• Basal Metabolic Rate (BMR) measures the energy expended while at rest in a neutral environment.
• BMR reflects the minimum calorific requirement to maintain vital bodily functions like breathing and circulation.
• In most calculations, 80% of the BMR reflects the body heat lost primarily through radiation.

Understanding metabolism is key in wellness and diet planning, ensuring that daily intake meets the body's energy needs.

Heat Transfer

Heat transfer is the movement of thermal energy from a hotter object to a cooler one. In humans, heat transfer plays an essential role in regulating body temperature. One specific type of heat transfer is radiation, where thermal energy is emitted in the form of electromagnetic waves. Various modes of heat transfer include:

• Conduction: Direct transfer of heat through molecules in contact.
• Convection: Transfer of heat through fluid motion, such as blood.
• Radiation: Transfer through electromagnetic waves without physical contact.

Radiation becomes significant in calculating heat loss for a person at rest, especially when considering basal metabolic rate and environmental conditions.

Surface Area Calculations

Surface area calculations are critical when determining how much heat a body can radiate. For humans, a larger surface area implies greater heat loss through radiation. The body surface area (BSA) is often estimated for physiological purposes and is utilized in many scientific and medical calculations.To approximate an individual's BSA for calculations, formulas such as the Dubois & Dubois method can be used:

• \( BSA (m^2) = 0.007184 \, \times (Height (cm)^{0.725}) \, \times (Weight (kg)^{0.425}) \)

Having accurate surface area measurements ensures valid calculations when applying the Stefan-Boltzmann Law to estimate heat radiated by the human body.